3.969 \(\int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx\)

Optimal. Leaf size=65 \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]

[Out]

(a*c^2*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + m)*x) + (b*c^2*(d*x)^(7 + m)*Sqrt[c*
x^2])/(d^7*(7 + m)*x)

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Rubi [A]  time = 0.0711002, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(c*x^2)^(5/2)*(a + b*x),x]

[Out]

(a*c^2*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + m)*x) + (b*c^2*(d*x)^(7 + m)*Sqrt[c*
x^2])/(d^7*(7 + m)*x)

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Rubi in Sympy [A]  time = 17.9647, size = 56, normalized size = 0.86 \[ \frac{a c^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 6}}{d^{6} x \left (m + 6\right )} + \frac{b c^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 7}}{d^{7} x \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**2)**(5/2)*(b*x+a),x)

[Out]

a*c**2*sqrt(c*x**2)*(d*x)**(m + 6)/(d**6*x*(m + 6)) + b*c**2*sqrt(c*x**2)*(d*x)*
*(m + 7)/(d**7*x*(m + 7))

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Mathematica [A]  time = 0.0361043, size = 39, normalized size = 0.6 \[ \frac{\left (c x^2\right )^{5/2} (d x)^m \left (\frac{a x^6}{m+6}+\frac{b x^7}{m+7}\right )}{x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(c*x^2)^(5/2)*(a + b*x),x]

[Out]

((d*x)^m*(c*x^2)^(5/2)*((a*x^6)/(6 + m) + (b*x^7)/(7 + m)))/x^5

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Maple [A]  time = 0.004, size = 40, normalized size = 0.6 \[{\frac{ \left ( bmx+am+6\,bx+7\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 6+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^2)^(5/2)*(b*x+a),x)

[Out]

x*(b*m*x+a*m+6*b*x+7*a)*(d*x)^m*(c*x^2)^(5/2)/(7+m)/(6+m)

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Maxima [A]  time = 1.3691, size = 53, normalized size = 0.82 \[ \frac{b c^{\frac{5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac{a c^{\frac{5}{2}} d^{m} x^{6} x^{m}}{m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)*(d*x)^m,x, algorithm="maxima")

[Out]

b*c^(5/2)*d^m*x^7*x^m/(m + 7) + a*c^(5/2)*d^m*x^6*x^m/(m + 6)

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Fricas [A]  time = 0.231621, size = 78, normalized size = 1.2 \[ \frac{{\left ({\left (b c^{2} m + 6 \, b c^{2}\right )} x^{6} +{\left (a c^{2} m + 7 \, a c^{2}\right )} x^{5}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 13 \, m + 42} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)*(d*x)^m,x, algorithm="fricas")

[Out]

((b*c^2*m + 6*b*c^2)*x^6 + (a*c^2*m + 7*a*c^2)*x^5)*sqrt(c*x^2)*(d*x)^m/(m^2 + 1
3*m + 42)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(c*x**2)**(5/2)*(b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)*(d*x)^m,x, algorithm="giac")

[Out]

Exception raised: TypeError